 8.1.8.1.1: In 16 write the linear system in matrix form.dx dt 3x 5y dydt 4x 8y
 8.1.8.1.2: In 16 write the linear system in matrix form.dx dt 4x 7y dy dt 5x
 8.1.8.1.3: In 16 write the linear system in matrix form.dx dt 3x 4y 9z dz dt 1...
 8.1.8.1.4: In 16 write the linear system in matrix form.dx dt x y dy dt x 2z d...
 8.1.8.1.5: In 16 write the linear system in matrix form.dx dt x y z t 1 dy dt ...
 8.1.8.1.6: In 16 write the linear system in matrix form. Dx dt 3x 4y e t sin 2...
 8.1.8.1.7: In 710 write the given system without the use of matrices.X 4 1 2 3...
 8.1.8.1.8: In 710 write the given system without the use of matrices.X 7 4 0 5...
 8.1.8.1.9: In 710 write the given system without the use of matrices.d dt x y ...
 8.1.8.1.10: In 710 write the given system without the use of matrices.d dt x y ...
 8.1.8.1.11: In 1116 verify that the vector X is a solution of the given system....
 8.1.8.1.12: In 1116 verify that the vector X is a solution of the given system....
 8.1.8.1.13: In 1116 verify that the vector X is a solution of the given system....
 8.1.8.1.14: In 1116 verify that the vector X is a solution of the given system....
 8.1.8.1.15: In 1116 verify that the vector X is a solution of the given system....
 8.1.8.1.16: In 1116 verify that the vector X is a solution of the given system....
 8.1.8.1.17: In 1720 the given vectors are solutions of a system X AX. Determine...
 8.1.8.1.18: In 1720 the given vectors are solutions of a system X AX. Determine...
 8.1.8.1.19: In 1720 the given vectors are solutions of a system X AX. Determine...
 8.1.8.1.20: In 1720 the given vectors are solutions of a system X AX. Determine...
 8.1.8.1.21: In 2124 verify that the vector Xp is a particular solution of the g...
 8.1.8.1.22: In 2124 verify that the vector Xp is a particular solution of the g...
 8.1.8.1.23: In 2124 verify that the vector Xp is a particular solution of the g...
 8.1.8.1.24: In 2124 verify that the vector Xp is a particular solution of the g...
 8.1.8.1.25: Prove that the general solution of on the interval (, ) is
 8.1.8.1.26: Prove that the general solution of on the interval (, ) is
Solutions for Chapter 8.1: Systems of Linear FirstOrder Differential Equations
Full solutions for Differential Equations with BoundaryValue Problems,  8th Edition
ISBN: 9781111827069
Solutions for Chapter 8.1: Systems of Linear FirstOrder Differential Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations with BoundaryValue Problems,, edition: 8. Since 26 problems in chapter 8.1: Systems of Linear FirstOrder Differential Equations have been answered, more than 21598 students have viewed full stepbystep solutions from this chapter. Differential Equations with BoundaryValue Problems, was written by and is associated to the ISBN: 9781111827069. Chapter 8.1: Systems of Linear FirstOrder Differential Equations includes 26 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Solvable system Ax = b.
The right side b is in the column space of A.